PHYSICAL REVIEW B 85, 165428 (2012)

Lagrange formalism of memory circuit elements: Classical and quantum formulations

Guy Z. Cohen,1,* Yuriy V. Pershin,2,† and Massimiliano Di Ventra1,‡1Department of Physics, University of California, San Diego, California 92093-0319, USA

2Department of Physics and Astronomy and University of South Carolina Nanocenter, University of South Carolina,Columbia, South Carolina 29208, USA

(Received 12 January 2012; revised manuscript received 14 March 2012; published 13 April 2012)

The general Lagrange-Euler formalism for the three memory circuit elements, namely, memristive,memcapacitive, and meminductive systems, is introduced. In addition, mutual meminductance, i.e., mutualinductance with a state depending on the past evolution of the system, is defined. The Lagrange-Euler formalismfor a general circuit network, the related work-energy theorem, and the generalized Joule’s first law are alsoobtained. Examples of this formalism applied to specific circuits are provided, and the corresponding Hamiltonianand its quantization for the case of nondissipative elements are discussed. The notion of memory quanta, thequantum excitations of the memory degrees of freedom, is presented. Specific examples are used to show thatthe coupling between these quanta and the well-known charge quanta can lead to a splitting of degenerate levelsand to other experimentally observable quantum effects.

DOI: 10.1103/PhysRevB.85.165428 PACS number(s): 73.23.−b, 45.20.Jj, 84.30.Bv, 03.65.Ca

I. INTRODUCTION

Circuit elements with memory, namely, memristive,1,2

memcapacitive, and meminductive3 systems, are attractingconsiderable attention in view of their application in diverseareas of science and technology, ranging from solid-statememories4–6 to neuromorphic circuits7–12 and understandingof biological processes.13,14 The general axiomatic definitionof memory elements considers any two fundamental circuitvariables, u(t) and y(t) [i.e., current I , charge q, voltage V ,or flux φ ≡ ∫ t

−∞ V (t ′)dt ′], whose relation, the response g,depends also on a set, x = {xi}, of n-state variables describingthe internal state of the system. These variables could be,e.g., the spin polarization of the sample15,16 or the positionof oxygen vacancies in a thin film.17 The resulting nth orderu-controlled memory circuit element is described by3

y(t) = g(x,u,t)u(t), (1)

x = f (x,u,t), (2)

where f is a continuous n-dimensional vector function. Itis assumed on physical grounds that, given an initial stateu(t = t0) at time t0, Eq. (2) admits a unique solution. If u

is the current and y(t) is the voltage, then Eqs. (1) and (2)define memory resistive (memristive) systems. In this caseg is the memristance (for memory resistance). In memorycapacitive (memcapacitive) systems, the charge is related to thevoltage so that g is the memcapacitance (memory capacitance),while in memory inductive (meminductive) systems the flux isrelated to the current with g, the meminductance (memoryinductance). These systems are characterized by a typical“pinched hysteretic loop” in their constitutive variables whensubject to a periodic input (with exceptions as discussed inRef. 18). Indeed, we have recently argued that essentially alltwo-terminal electronic devices based on memory materialsand systems, when subject to time-dependent perturbations,behave simply as, or as a combination of, memristors,memcapacitors, and meminductors.3,19 This unifying descrip-tion is a source of inspiration for novel digital and analog

applications18,20,21 and allows us to bridge apparently differentareas of research.3

However, despite the wealth of applications and new ideasthese concepts have generated, it is nonetheless important tostress that so far these memory elements have been discussedonly within their classical circuit theory definition, with quan-tum mechanics entering at best in the microscopic parametersthat determine the state variables responsible for memory.3,17,22

However, it seems that these features are common at thenanoscale where the dynamical properties of electrons andions are likely to depend on the history of the system, at leastwithin certain time scales.23,24 Mindful of the trend towardextreme miniaturization of devices of all sorts, it is thus naturalto ask whether true quantum effects can be associated withthe memory of these systems and which phenomena couldemerge from the quantization of memory elements. Of course,examples of memory effects in quantum phenomena can befound in the specialized literature (see, e.g., Ref. 25). Hereinstead, we want to provide a general framework of study ofthe quantum excitations (memory quanta) associated to generaldegrees of freedom that lead to memory in these systems.

We then first introduce the general Lagrange-Euler formal-ism for these systems. This is the nontrivial extension of thecorresponding formalism for the “standard” circuit elements.Since it is well known that the Lagrangian formulation ofcircuit elements offers great advantages in the analysis ofcomplex circuits,26 we expect that this generalization wouldbe of great value in itself. Moreover, our work extendsprevious studies related to the formulation of Lagrangeand Routh equations for nonlinear circuits involving idealmemristors27 and to the port-Hamiltonian modeling for thecase of memristive components.28 In the present contextour work also sheds light on the general relation betweenthe internal degrees of freedom that lead to memory andthe constitutive variables—the charge, current, voltage, andflux—that define the different elements. Along the way wealso define mutual meminductors, namely, mutual inductorswith memory, which add additional flexibility and hence newfunctionalities to the field of memory elements.

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COHEN, PERSHIN, AND DI VENTRA PHYSICAL REVIEW B 85, 165428 (2012)

We finally proceed to quantize the corresponding equationsin the standard way. This leads us to consider the memoryexcitations of these systems. In this paper we consider onlythe quantization of nondissipative elements, and we will devotea subsequent paper to the discussion of quantum effects indissipative memory elements. We will provide examples ofapplications of the Lagrangian formalism to selected cases anddiscuss experimental conditions under which these memoryquanta could be detected.

This paper is organized as follows. In Sec. II we introducea general scheme of the approach. Section III is dedicatedto the Lagrangian formulation of memristive systems, whileSecs. IV and V deal with memcapacitive and meminduc-tive systems, respectively. We then show how to write theLagrangian (Sec. VI) and Hamiltonian (Sec. VII) of a circuit ofmemory elements and also give the work-energy theorem andgeneralized Joule’s first law for such a circuit. We introduce theconcept of memory quanta in Sec. VIII, focusing on specificexamples. Finally, in Sec. IX we report our conclusions.

II. LAGRANGE APPROACH

In the Lagrange formalism, each memory circuit elementis associated with m + 1 degrees of freedom [one related to acircuit variable (q or φ) and m to its internal state (generalizedcoordinates, yj , j = 1,...,m)]. For convenience, we define twomultivariate vectors:

Y q = (q,y1,...,ym), (3)

Yφ = (φ,y1,...,ym). (4)

We note that there are two (in some cases, however, one)internal state variables xi [entering Eqs. (1) and (2)] for eachyj . Quite generally then x = {y,y} [with y here not to beconfused with the output variable y(t) in Eq. (1)].

A model of any particular memory circuit element consistsof three components: the kinetic energy T , the potential energyU , and the dissipation potential H. The m + 1 Lagrangeequations of motion are given by

d

dt

∂L∂Y α

j

− ∂L∂Y α

j

= QYαj, (5)

where L = T − U is the Lagrangian; α is q or φ; j = 0,...,m;and the generalized dissipation force QYα

jis defined as

QYαj

= − ∂H∂Y α

j

. (6)

While, generally, models of different memory circuit elementsinvolve similar terms related to internal degrees of freedom,the contribution from the circuit variable q or φ is specific foreach type of memory circuit element as presented in Table I.

The kinetic energy T may have a contribution describingthe dynamics of internal degrees of freedom and a specificcontribution according to Table I. The contribution frominternal degrees of freedom, T , can be written using symmetryarguments. First of all, since dissipative effects are not includedin the kinetic energy, it is time-reversal invariant, and onlyeven powers of yi can exist. In order for the transformation tocanonical momenta to be invertible, however, we must leave

TABLE I. General scheme of Lagrange description of memorycircuit elements. Specific contributions listed in the columns T , U ,and H are given by Eqs. (16), (22), (34), (39), (50), and (57).

System type Variables T U H

V -controlled memristive system Y q HMV

I -controlled memristive system Y φ HMI

V -controlled memcapacitive system Y q UCV

q-controlled memcapacitive system Y φ T Cq

φ-controlled meminductive system Y q T Lφ

I -controlled meminductive system Y φ ULI

only quadratic terms, and we find T = ∑ij cij yi yj . This form,

being symmetric, can be diagonalized to give

T = T + T βu =

∑i

ci˙y

2i

2+ T β

u , (7)

where ci are real positive numbers to be determined micro-scopically, and T

βu is the specific contribution, if it exists (see

Table I), to the kinetic energy for the u-controlled memorycircuit element, β = M,C, or L.

The potential energy U and dissipative potential H alsoinclude a specific contribution from Table I and contributionsfrom internal degrees of freedom U and H:

U = U (y,u,t) + Uβu , (8)

H = H(y,y,u,t) + Hβu , (9)

where y = {yi}. It is important to consider the control variableu as an independent parameter that can be replaced by an(output) circuit variable [using, e.g., Eq. (1)] only in the finalequations of motion.

III. MEMRISTIVE SYSTEMS

There are two types of memristive systems: voltage-controlled and current-controlled ones.2 From Eqs. (1) and(2), we define voltage-controlled memristive systems by theequations

IM (t) = R−1(x,VM,t)VM (t), (10)

x = f (x,VM,t), (11)

where VM (t) and IM (t) = q(t) denote the voltage and currentacross the device, and R is the memristance and its inverseis the memductance (for memory conductance). A current-controlled memristive system is such that the resistance andthe dynamics of state variables depend on the current:2,18

VM (t) = R(x,IM,t)IM (t), (12)

x = f (x,IM,t). (13)

At this point we note that the above equations have beenintroduced to define a wide class of systems collectively calledmemristive,2 while the name memristor1 has been assignedto the ideal case of these equations, when R depends onlyon the voltage (or current) history. Although some authorsuse the term memristor to represent any system that satisfies

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(a) (b)

FIG. 1. (a) Schematic of a voltage-controlled memristive systemconnected to a time-dependent voltage source. (b) Schematic of acurrent-controlled memristive system connected to a time-dependentcurrent source.

Eqs. (10) and (11) or (12) and (13), we reserve this termfor the ideal case only.1 (We will also see in Sec. VI C thatsuch systems, like ideal memcapacitors and meminductors,require special care in the Lagrangian formulation.) We alsonote that, often, current-controlled memristive systems canbe redefined as voltage-controlled ones and vice versa.18

In addition, according to Thevenin’s theorem,29,30 a voltagesource V (t) in series with a resistance R is equivalent toa current source I (t) = V (t)/R with the same resistance inparallel. We could then choose to work with either one of thesecases. However, for completeness, in the following we willpresent the Lagrangian formalism for both voltage-controlledand current-controlled memristive systems.

A. Voltage-controlled systems

We consider a voltage-controlled memristive system con-nected to a time-dependent voltage source V (t) as shown inFig. 1(a). In addition to the term due to internal degrees offreedom discussed in Sec. II, the total potential energy containsthe usual contribution from the battery −qV (t), with q beingthe charge that flows in the circuit. There are many mechanismsfor potential energy arising from the state variables which areaffected by the applied bias—an example of this is the changeof state due to electromigration (see, e.g., Ref. 24). The totalpotential energy is thus given by

U = U (y,VM,t) − qV (t), (14)

so that the Lagrangian is

L = T − U =∑

i

ci y2i

2− U (y,VM,t) + qV (t). (15)

Here, VM is considered as an independent parameter.As shown in Table I, the dissipation potential of voltage-

controlled memristive systems includes a circuit variablecontribution HM

V . We write it similarly to the well-knownRayleigh’s “dissipation potential” (for a constant value resis-tor) of the type H = Rq2/2, which gives rise to a “dissipationforce” Qq = −∇qH = −Rq.31 Specifically, we will use

HMV = R(y,VM,t)q2

2. (16)

At this point we stress that the memristance (as well as thememcapacitance and meminductance we will discuss later)may also depend on generalized velocities, y, which are alsoincluded into x. This would simply modify the Lagrangeequations of motion without changing the overall formalism.To simplify the notation, however, we will not include this

dependence explicitly here, and we give an explicit exampleof this case in Sec. V B.

For the total dissipation potential we write

H = R(y,VM,t)q2

2+ H(y,y,VM,t), (17)

where the last term is to be determined phenomenologicallyor from a microscopic theory.

It is straightforward to show that the equation of motion (5)for Y

q

0 = q can be written as

V (t) ≡ VM (t) = R(y,VM,t) q. (18)

This equation is of the type of Eq. (10). The correspondingequations of motion for the state variables x are

ci yi + ∂H(y,y,VM,t)

∂yi

+ ∂U (y,VM,t)

∂yi

= 0, (19)

which show explicitly two possible physical origins of memris-tance due to a dissipative component and/or a potential-energycomponent.

Equation (19) can be rewritten as two first-order differentialequations of the form of Eq. (11) considering both yi and yi

as internal state variables. Moreover, in the final equations wecan substitute VM by its expression in terms of the currentq. For this purpose, Eq. (10) can be solved with respect toVM . The same final procedure can also be used in the case ofmemcapacitive and meminductive systems considered below.

B. Current-controlled systems

As a simple example of a closed circuit with a current-controlled memristive system, we consider a source of currentI (t) connected to a memristive system [Fig. 1(b)]. Here, asindicated in Eqs. (12) and (13), the output circuit variable isthe voltage across the memristive system, φ = VM (t). Thatis why we use Yφ set of variables in this case. The kineticenergy, potential energy, and total dissipation potential in theLagrangian formalism are now

T = 1

2

∑i

ci y2i , (20)

U = U (y,IM,t) − φI (t), (21)

H = HMI + H = φ2

2R(y,IM,t)+ H(y,y,IM,t), (22)

where −φI (t) is the battery term. Although not a necessarystep (if their values are known), R(y,IM,t), U (y,IM,t),H(y,y,IM,t) can be obtained from R(y,VM,t), U (y,VM,t),and H(y,y,VM,t), correspondingly. However, the solutionmay be multiple valued in IM so that R(y,y,IM,t) may havemultiple branches with the correct choice of branch dependingon the history of the memristive system.

The equation of motion (EOM) for φ follows from Eq. (5)for Y

φ

0 = φ, taking into account Eqs. (20)–(22), leading to

φ = R(y,IM,t)IM, (23)

which is just Eq. (12). The EOM for x is similarly found fromEq. (5), resulting in Eq. (19) except for the substitution of VM

by IM .

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C. Example

Here we provide a specific physical example to clarifyboth the formalism and the different terms that appearin Eqs. (18) and (19). For this we consider a thermistor,namely, a temperature-dependent resistor. The memristivemodel of thermistor2,18 utilizes a single internal state variable,the absolute temperature of the thermistor, x = y = Ttherm,and can be formulated as a first-order voltage-controlledmemristive system.18 Mathematically, the Lagrangian modelof the thermistor involves the following kinetic and potentialenergies and dissipation potentials:

T = 0, (24)

U = 0, (25)

HMV = R(y)q2

2, (26)

H = y

[1

2Chy − V 2

M

R(y)− (Tenv − y)δ

], (27)

where R(y) = R0eβ(1/y−1/T0) is the temperature-dependent

resistance, R0 denotes the resistance at a certain temperatureT0, β is a material-specific constant, Ch is the heat capacitance,δ is the dissipation constant of the thermistor,2 and Tenv is thebackground (environment) temperature.

Using Eq. (5) for a circuit consisting of a thermistorconnected to a voltage source V (t) [see Fig. 1(a)], we recoverthe equations of the memristive model of the thermistor:18

I = [R0eβ(1/y−1/T0)]−1VM, (28)

Ch

dy

dt= [R0e

β(1/y−1/T0)]−1V 2M + (Tenv − y)δ. (29)

Note that although other forms of potential- and kinetic-energyterms could produce the same Eqs. (28) and (29) this particularone also satisfies the Joule’s first law discussed in Sec. VII C.This puts severe constraints on the choice of Lagrangian.

IV. MEMCAPACITIVE SYSTEMS

We now consider memcapacitive systems3 (Fig. 2),which—unlike memristive systems—store also energy. In par-ticular, voltage-controlled memcapacitive systems are definedby Eqs. (1) and (2), with u being the voltage, VC(t), acrossthe memcapacitive system, and y(t) being the charge, qC(t),

(a) (b)

FIG. 2. (a) Schematic of a voltage-controlled memcapacitivesystem connected to a time-dependent voltage source. (b) Schematicof a charge-controlled memcapacitive system connected to a time-dependent current source.

stored in the device, leading to

qC(t) = C(x,VC,t)VC(t), (30)

x = f (x,VC,t), (31)

where C is the memcapacitance. As in memristive systems,the above equations define a large class of systems, with idealmemcapacitors being those for which the memcapacitancedepends only on the voltage history (or, for charge-controlledmemcapacitive systems, only on the charge history).3 Inaddition, it is often important to consider the energy addedto or removed from a memcapacitive system, namely, thequantity UC = ∫ t

t0VC(τ )I (τ )dτ which helps understanding of

whether a memcapacitive system is nondissipative, dissipative,or active.3 Of these, the nondissipative and/or dissipativememcapacitive systems are the most interesting for potentialapplications, and we will therefore focus here on these casesonly.

A charge-controlled memcapacitive system is defined bythe set of equations3

VC = C−1(x,qC,t)qC(t), (32)

x = f (x,qC,t). (33)

A. Voltage-controlled systems

The Lagrange model of voltage-controlled memcapacitivesystems is based on the Y q set of variables (see Table I).The specific contribution from the q degree of freedom to thepotential energy is

UCV = q2

2C(y,VC,t). (34)

Taking into account a voltage source connected to the system[Fig. 2(a)], the total potential energy is written as

U = q2

2C(y,VC,t)+ U (y,VC,t) − qV (t). (35)

Consequently, the Lagrangian is given by

L = T − U =∑

i

ci y2i

2− q2

2C(y,VC,t)

− U (y,VC,t) + qV (t). (36)

The dissipative potential contains only the internal statevariables contribution H(y,y,VC,t).

The Lagrange EOMs for voltage-controlled memcapacitivesystems have the form

q(t)

C(y,VC,t)= V (t) ≡ VC(t), (37)

ci yi + ∂H(y,y,VC,t)

∂yi

+ ∂U (y,VC,t)

∂yi

− V 2C

2

∂C(y,VC,t)

∂yi

= 0, (38)

where in writing the last term in Eq. (38) we have madeuse of Eq. (37). Its clear that Eqs. (37) and (38) are of theform of Eqs. (30) and (31) In fact, Eq. (38) clearly shows

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that the memory may arise from both a conservative potentialcontribution as well as a dissipative one.

Equation (38) describes an effective dynamical system and,together with Eq. (30), tells us that, in the presence of a periodicinput of frequency ω, charge dynamics can be out of phase withthe voltage across the memcapacitive system. Indeed, theremight be a delay in response of the internal state variablesto the applied voltage leading to the above-mentioned effect.Experimentally, it can be seen as a pinched hysteresis loop inthe q-VC plane.3,18

B. Charge-controlled systems

We consider a circuit consisting of a current source and acurrent-controlled memcapacitive system [Fig. 2(b)]. Here, asseen in Eqs. (32) and (33), the circuit variable is the voltageacross the memcapacitive system, φ = VC(t), instead of thecurrent through it, q = IC(t), as in voltage-controlled systems[c.f. Eqs. (30) and (31)]. Consequently, our analysis should bebased on the Yφ set (Table I). The kinetic energy, potentialenergy, and total dissipation potential in the Lagrangianformalism are now

T = T + T Cq = 1

2

∑i

ci y2i + 1

2C(y,qC,t)φ2, (39)

U = U (y,qC,t) − φI (t), (40)

H = H(y,y,qC,t). (41)

The EOM for φ is derived by applying Eq. (5) to Eqs. (39)–(41), leading to Eq. (32). The EOM for x is similarly obtainedand results in Eq. (38) except for the substitution of VC by qC .

C. Example

As an example of a voltage-controlled memcapacitive sys-tem we consider a parallel-plate capacitor with an elasticallysuspended upper plate and a fixed lower plate.18 When a chargeis added to the plate, the separation between plates changes asoppositely charged plates experience an attractive interaction.The internal degree of freedom of the elastic memcapacitivesystem is the position of the upper plate y measured froman equilibrium uncharged plate separation, d0. The Lagrangemodel of elastic memcapacitive system connected to a voltagesource consists of the following kinetic and potential energiesand dissipation potentials:

T = my2

2, (42)

U = UCq + U = q2

2C(y)+ ky2

2− qV (t), (43)

H = H = γmy2

2, (44)

supplemented by the expression for the memcapacitance,C(y) = C0/(1 + y/d0). Here, m is the mass of the upperplate, γ is a damping coefficient representing dissipation ofthe elastic oscillations, k is the spring constant, C0 = εS/d0 isthe equilibrium value of capacitance, S is the plate area, and ε

the permittivity of the medium.The first equation of motion is of the form of Eq. (37); the

second, Eq. (31), is obtained by substituting Eqs. (42)–(44) into

(a) (b)

FIG. 3. (a) Schematic of a flux-controlled meminductive systemconnected to a time-dependent voltage source. (b) Schematic ofa current-controlled meminductive system connected to a time-dependent current source.

Eq. (38). Explicitly, we obtain the classical harmonic-oscillatorequation including damping and driving terms:

d2y

dt2+ γ

dy

dt+ ω2

0y + V 2C

2md0

C0

(1 + y/d0)2= 0. (45)

Here, ω0 = √k/m. To emphasize the similarity of Eq. (45)

with Eq. (31) we note that Eq. (45) can be written as twofirst-order differential equations and the internal state variablesare x1 = y and x2 = y.

V. MEMINDUCTIVE SYSTEMS

Let us finally consider meminductive systems3 (Fig. 3). Aflux-controlled meminductive system satisfies the relations3

IL = L−1(x,φL,t)φL(t), (46)

x = f (x,φL,t), (47)

with L−1 being the inverse meminductance. A current-controlled meminductive system is defined by the set ofequations3

φL = L(x,IL,t)IL(t), (48)

x = f (x,IL,t). (49)

As in the case of memcapacitive systems, meminductiveelements may represent nondissipative, dissipative, or activedevices. We are interested only in the first two types since theyare the most important for technological applications.

A. Flux-controlled systems

Consider a circuit composed of a voltage source connectedto a meminductive system as in Fig. 3(a). The circuit degreeof freedom q in flux-controlled meminductive systems istaken into account by the following contribution to the kineticenergy:

T Lφ = L(y,φL,t)q2

2. (50)

The contributions to T , U , and H from internal state degreesof freedom are written in the general form [Eqs. (7)–(9)].Consequently, taking also a voltage source in Fig. 3(a) intoaccount, the Lagrangian and dissipative potential are writtenas

L =∑

i

ci y2i

2+ L(y,φL,t)

q2

2− U (y,φL,t) + qV (t), (51)

H = H(y,y,φL,t). (52)

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The EOM for the q degree of freedom is

d[L(y,φL,t)q]

dt= V (t) ≡ VL(t). (53)

Integrating this equation in time assuming that φ(t = −∞) =0 we find

L(y,φL,t)q =∫ t

−∞dt ′ VL(t ′) = φL(t), (54)

which is Eq. (46).The EOMs for the state variables are written as

ci yi + ∂H (y,y,φL,t)

∂yi

+ ∂U (y,φL,t)

∂yi

− φ2L

2L2

∂L(y,φL,t)

∂yi

= 0, (55)

which again, since x = {y,y}, can be written in the form ofEq. (47).

B. Current-controlled systems

When one considers circuits involving current-controlledmeminductive systems (and current sources instead of voltagesources), one should employ the Yφ set of variables (seeTable I). Let us then consider a simple circuit composedof a current source directly connected to a meminductivesystem [Fig. 3(b)]. The contribution from the circuit degreeof freedom φ comes from the potential-energy UL

I term. Thekinetic energy, potential energy, and total dissipation potentialin the Lagrangian formalism are now

T = T = 1

2

∑i

ci y2i , (56)

U = U + ULI − φI = U (y,IL,t) + φ2

2L(y,IL,t)− φI (t),

(57)

H = H(y,y,IL,t). (58)

The EOM for φ is the same as Eq. (48). The EOMs for yi

are similarly derived and result in Eq. (55) except for thesubstitution of φL by IL.

C. Example

We here provide an instructive example of an effectivememinductive system consisting of an LCR contour induc-tively coupled to an inductor (Fig. 4). In this scheme, the twoinductors L1 and L2 interact with each other magnetically.From the point of view of the voltage source V (t), thetotal system can be seen as a second-order flux-controlledmeminductive system described by the general Eqs. (46) and(47). The charges on the capacitor C and the current throughthe inductor L2 play the role of internal state variables. It isconvenient to select y = qC . Consequently, IL2 = y.

We start by considering the circuit presented in Fig. 4 usingthe Lagrange formalism for usual circuit elements. The circuit

FIG. 4. Flux-controlled meminductive system based on the in-ductive coupling of a coil L1 with a LCR contour. Here, the mutualinductance M is equal to k

√L1L2, where 0 � k � 1 is the coupling

coefficient.

is described by

T = 1

2L1q

2 + 1

2L2y

2 + Myq, (59)

U = y2

2C− qV (t), (60)

Hq = 0, (61)

Hx = Ry2

2. (62)

The EOMs for x and q are then found to be

L1q + My − V (t) = 0, (63)(L2 − M2

L1

)y + M

L1V (t) + y

C+ Ry = 0. (64)

One can easily verify that Eqs. (63) and (64) describe theelectric circuit from Fig. 4.

Next, integrating Eq. (63) (the constant of integration istaken to be zero), we can rewrite it in the form

φ(t) = L1φ(t)

φ(t) − Myq(t) ≡ L[y,φ(t)]q(t), (65)

which shows that the meminductance L depends on thegeneralized velocity x = y.

D. Mutual meminductance

After the generalization of self-inductance to meminduc-tance, one wonders if mutual inductance can be generalized tomemory situations as well. We consider two coupled inductors,as in Fig. 4, but now assume the mutual inductance to havememory. Here, we want to describe that part of the memorythat cannot be included in two (self-)meminductive systems.This memory can be stored in the medium between theinductors with a state affected by the two magnetic fluxes of theinductors. It could also be stored in the geometry of the systemby having, e.g., two elastic coils that can either attract or repeleach other. Since the memory mechanism does not belongsolely to one inductor, the relation M = k

√L1L2, applicable

to mutual inductance of two coils, does not apply for mutualmeminductance: k is not generally a constant independent ofL1, L2, x, and possibly some other parameters.

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LAGRANGE FORMALISM OF MEMORY CIRCUIT . . . PHYSICAL REVIEW B 85, 165428 (2012)

FIG. 5. Symbol for a mutual meminductor.

In analogy with Eqs. (46) and (47), we then define a flux-controlled mutual meminductive system via the following setof equations:

q1 = M−1(x,φM1,φM2,t)φM2, (66)

q2 = M−1(x,φM1,φM2,t)φM1, (67)

x = f (x,φM1,φM2,t), (68)

where M(x,φM1,φM2,t) is the mutual meminductance, φM1 isthe magnetic flux defined by φM1 = ∫ t

−∞ VM1(t ′)dt ′, VM1(t) isthe voltage on the first inductor [φM2 and V2M (t) are similarlydefined], and q1 and q2 are the currents in the first and secondinductors, respectively. The circuit symbol we propose for thiselement is shown in Fig. 5.

Regarding the Lagrangian formulation, the additions to thekinetic energy, potential energy, and dissipation potential as aresult of introducing this memory element are

T = 1

2

∑i

ci x2i + M(x,φM1,φM2,t)q1q2, (69)

U = U (x,φM1,φM2,t), (70)

H = Hx(x,x,φM1,φM2,t). (71)

The corresponding current-controlled mutual meminductivesystems are instead defined by the set of equations

φM1 = M(x,IM1,IM2,t)IM2, (72)

φM2 = M(x,IM1,IM2,t)IM1, (73)

x = f (x,IM1,IM2,t), (74)

where IMi is the current in the ith inductor and M(x,IM1,IM2,t)is the mutual inductance that can be obtained by plugging inφM1(IM2) and φM2(IM1) in M(x,φM1,φM2,t). The Lagrangianformulation of this system is given by

T = 1

2

∑i

ci x2i , (75)

U = U (x,MIM2,MIM1,t) + φM1φM2

M(x,IM1,IM2,t), (76)

H = Hx(x,x,MIM2,MIM1,t), (77)

where both U and Hx are the same functions as defined in theabove flux-controlled case.

VI. LAGRANGIAN OF A GENERAL CIRCUIT

We now have all the ingredients to write down theLagrangian for a general circuit network composed of anarbitrary combination of memristive, memcapacitive, andmeminductive systems and their standard counterparts. These

circuits may be powered by an arbitrary set of voltage sourcesVk(t) (Sec. VI A), for which the fluxes φk(t) are defined as,e.g., in Eq. (54), or by a set of current sources Ik(t) (Sec. VI B).When both voltage sources and current sources are present, onecan convert the latter to the former using Thevenin’s theorem orthe former to the latter using Norton’s theorem,32 thus ensuringonly one type of power source is present. Below we brieflyoutline the recipe to write the Lagrangian of a general circuitfor both cases.

A. Circuits with voltage sources

A general electronic circuit powered by voltage sources canbe described as a combination of l indivisible loops, i.e., onesthat do not contain internal loops. Within the j th (j = 1, . . . ,l)loop one should consider the charge qj as the circuit variableand take into account generalized coordinates of elementsinvolved in this loop. For simplicity, we rename the generalizedcoordinates for the whole circuit as yi (i = 1, . . . ,k).

The current in each branch of the circuit is the sum ofcontributions from indivisible loops it belongs to. Using thisfact, we can write the Lagrangian for each element in thebranch. The element’s Lagrangian is taken in the voltage-controlled form for memristive and memcapacitive systemsand in the flux-controlled form for meminductive ones.

The sum of the Lagrangians of individual elements of thecircuit gives the circuit’s Lagrangian, while the sum of thedissipation potentials gives the circuit’s dissipation potential.The circuit’s Lagrangian and dissipation potential depend onqj , qj , yi , and yi and result in l + k EOMs. The EOM obtainedfor qj gives Kirchhoff’s voltage law (KVL) for the j th loopbecause of the linearity of the Euler-Lagrange equations andbecause each component in the loop was shown above to givethe correct voltage term. Kirchhoff’s current law (KCL), onthe other hand, is automatically satisfied by the choice of loopcurrent variables.

B. Circuits with current sources

When a circuit is powered by current sources, the circuitvariable is the flux φj in the j th (j = 1, . . . ,l) junction, whileφj is the electric potential at the junction. Using this definition,the flux or voltage across each element in the network canbe found via the difference of the fluxes or potentials inthe junctions at its ends, enabling one to write the element’sLagrangian and dissipation potential in the current-controlledformalism for memristive and meminductive systems andin the charge-controlled formalism for memcapacitive ones.As in the voltage-controlled case, the circuit’s Lagrangianor dissipation potential is the sum of the circuit element’sLagrangians or dissipation potentials, respectively.

If we denote again the internal degrees of freedom ofthe whole circuit as yi (i = 1, . . . ,k), we have a circuit’sLagrangian and dissipation potential that depend on φj , φj ,yi , and yi and result in l + k EOMs. The EOM obtained forφj gives KCL for the j th junction due to the linearity of theEuler-Lagrange equations and because each element endingon the junction was shown above to give the correct currentterm. KVL, on the other hand, is automatically satisfied by thechoice of junction potential variables.

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FIG. 6. (Color online) Schematic of flux-controlled inductively-coupled charging circuits with a dc voltage source

This formalism has a complementary nature and can beviewed as the dual formalism to that for circuits with voltagesources. This conclusion will be reinforced in Sec. VII, wherewe will show the canonically conjugate momenta of thevoltage-controlled and current-controlled formalisms to befluxes and charges, respectively.

C. Lagrangian multipliers

When a circuit, voltage-controlled or current-controlled,has additional constraints—missing from the EOMs—relatingstate variables to circuit variables, the form of their dependenceshould be added to the Lagrangian. This is achieved by themethod of Lagrange multipliers. In particular, the Lagrangemultipliers are convenient for describing ideal memory circuitelements such as the ideal memristor,18 in which the statevariable y equals the charge q flowing through the device.Examples for such constraints appear in Sec. VI D.

D. Examples

Consider two inductively coupled RC circuits as shownin Fig. 6. Initially, both capacitors are not charged and thereare no currents in the circuits. The circuits are coupled viamutual inductance M , which results in periodic charging anddischarging of the right-hand side capacitor as will be seenbelow. We take a memristor with R(y) (y = q) qualitativelysimilar to the one fitted recently to experiments on TiO2 thinfilms,17 namely, of the form

R(y) = Ron + Roff − Ron

1 + y2/q20

, (78)

where Ron, Roff , and q0 are parameters defined for eachmemristor. The resistance is seen to decrease from Roff toRon as charge flows through the memristor. Denoting the leftand right loop charges as q1 and q2, respectively, and applyingthe results of Secs. III–V, we obtain the following Lagrangianand dissipation potentials for the network:

L = L

2

(q2

1 + q22

) − Mq1q2 + q1V1 − q21 + q2

2

2C− λ(y − q1),

(79)

Hq = 1

2R(y)q2

1 + 1

2Rq2

2 , (80)

Hx = 0, (81)

where λ is the Lagrange multiplier corresponding to the circuitholonomic constraint, y = q1 [a different constraint wouldlead to a different corresponding term in Eq. (79)]. These

0 2. 10 7 4. 10 7 6. 10 7 8. 10 70

5. 10 13

1. 10 12

1.5 10 12

2. 10 12

2.5 10 12

3. 10 12

3.5 10 12

t sec

q 1C

oul

0 2. 10 7 4. 10 7 6. 10 7 8. 10 72. 10 13

1. 10 13

0

1. 10 13

2. 10 13

t sec

q 2C

oul

FIG. 7. Graphs of q1(t) (top) and q2(t) (bottom) for the systemdescribed in Fig. 6. The solid line corresponds to the behavior withthe memristor and the dashed line to the behavior with the memristorreplaced by a normal resistor of resistance Roff . The parameters usedwere V1 = 1 V, R = 10 k�, Roff = 100 k�, Ron = 100 �, C = 3 pF,M = L = 0.3 mH, q0 = 10−12 C. The initial conditions were set tono charge or current in any of the circuit elements.

expressions lead to the following EOMs for the total system:

Lq1 − Mq2 + R(q1)q1 + q1

C− V1 = 0, (82)

Lq2 − Mq1 + Rq2 + q2

C= 0, (83)

where the EOM for y, giving λ = 0, and the EOM for λ,giving y = q1, were substituted. The solution of these EOMsfor certain values of the parameters is shown in Fig. 7. Wenote that the insertion of the memristor produces an almostconstant current instead of an exponentially decreasing one inthe left loop of the circuit. The stabilization of the current isachieved by the decline in the characteristic charging timeR(q1)C as the capacitor is charged. The memristor alsomodifies the exchange of energy between the two circuits,giving pronounced oscillations in the charge of the right loop ofthe circuit, which are absent when the memristor is substitutedwith a normal resistor.

As a second example, consider the circuit shown inFig. 8. The circuit consists of a 3 × 3 network of memristorsconnected to a voltage source. Each memristor has resistanceR(yi), where yi is the cumulative charge that flows through thememristor. The indivisible loops’ charges are denoted by qi .The EOMs can be readily obtained from the Lagrangian anddissipation potential of the circuit that read

L = qV (t) − λ1(y1 − q1) − λ2(y2 − q2) − λ3(y3 − q0 + q1)

− λ4(y4 − q1 + q2) − λ5(y5 − q2) − λ6(y6 − q3 + q1)

− λ7(y7 − q4 + q2) − λ8(y8 − q0 + q3)

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LAGRANGE FORMALISM OF MEMORY CIRCUIT . . . PHYSICAL REVIEW B 85, 165428 (2012)

FIG. 8. Schematic of a voltage-controlled 3 × 3 network ofmemristive systems with a voltage source.

− λ9(y9 − q3 + q4) − λ10(y10 − q4)

− λ11(y11 − q0 + q3) − λ12(y12 − q0 + q4), (84)

H = 12

[R(y1)q2

1 + R(y2)q22 + R(y3)(q0 − q1)2

+R(y4)(q1 − q2)2 + R(y5)q22 + R(y6)(q3 − q1)2

+R(y7)(q4 − q2)2 + R(y8)(q0 − q3)2 + R(y9)(q3 − q4)2

+R(y10)q24 + R(y11)(q0 − q3)2 + R(y12)(q0 − q4)2],

(85)

where λi are the Lagrange multipliers. These two functionscan be easily generalized for the case of a N × M networkof different memristors, greatly facilitating the attainment ofthe EOMs, the solution of which can be used to solve, e.g.,optimization problems such as mazes in a massively parallelway.21

VII. HAMILTON FORMALISM

The counterpart of the Lagrange formalism is the Hamiltonone, which is also generally the starting point for quantization.For nondissipative systems, one can easily transform theLagrangian to the Hamiltonian. In the presence of dissipationinstead, this task requires particular care.

Dissipation is the result of the tracing out of certain degreesof freedom resulting in an effective (reduced) description ofthe system of interest in interaction with these degrees offreedom. However, the microscopic procedure of tracing outthese degrees of freedom is most of the time difficult to carryout exactly, and dissipation is then introduced with physicallyplausible ad hoc strategies.

There are several ways to add dissipation at the level ofcircuit Hamiltonians which range from complex Lagrangians(resulting in complex Hamiltonians)33 to the addition oflinear dissipative elements modeled by an infinite networkof capacitors and inductors (see, e.g., Ref. 34). Since the dis-cussion of dissipation in Hamiltonian dynamics would requirean extensive treatment by itself, here we limit our analysisto nondissipative systems and (except for the work-energy

theorem discussed below) leave the Hamiltonian formalism ofdissipative memory elements for a future publication.

A. Canonically conjugate momenta

Consider a nondissipative network of memory elements.In order to write the Hamiltonian, we need to determinethe momenta pj canonically conjugate to the variables qj .These momenta are defined for circuits of voltage-controlledelements by

pj ≡ ∂L∂qj

, (86)

with the same definition for circuits of current-controlledelements except for qj being replaced by φj . Looking atthe expressions for the Lagrangians of the memory elementsdiscussed above, one easily finds the physical meaning ofpj . In voltage-controlled circuits pj is the total of the fluxesgenerated by the inductors in the j th loop, while in current-controlled circuits it is the charge in the j th junction. Inaddition, if we define the canonically conjugate momentumto the internal degree of freedom yi as zi , we readily find thatfor both voltage-controlled and current-controlled circuits

zi ≡ ∂L∂yi

= ci yi . (87)

The Hamiltonian equations for voltage-controlled circuitsthen read

qj = ∂H

∂pj

, (88)

pj = −∂H

∂qj

, (89)

with qj replaced by φj for current-controlled circuits.Using the results for the canonically conjugate momenta,

we see that for voltage-controlled circuits Eq. (88) gives thecurrent in the j th loop in terms of the magnetic flux in theinductors in each loop, while Eq. (89) gives the change in themagnetic flux in the inductors in the j th loop in terms of thecharges in the loops. For current-controlled circuits, on theother hand, Eq. (88) gives the potential in the j th junctionin terms of the charges in the circuit junctions, and Eq. (89)gives the current flowing into this junction in terms of thefluxes in the circuit junctions. The EOMs obtained here—whilerepresenting the same physics—are distinctly different fromthe ones in the Lagrangian formalism, and therein lies theirvalue.

With these results in mind, the Hamiltonian is defined asthe Legendre transformation of the Lagrangian, namely,

H =∑

j

pj qj +∑

i

zi yi − L (90)

for voltage-controlled circuits, and with qj substituted by φj

for current-controlled circuits. Since the kinetic energy in bothcases is quadratic in qj (φj for current-controlled circuits), itis easy to see that Eq. (90) reduces to H = T + U .

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B. Work-energy theorem

If we consider an arbitrary circuit with a number of voltagesources Vkj (of the kth voltage source in the j th loop), we candefine the work done by these sources, at any given time in aninterval of time dt , on infinitesimal charges dqj in each of theindivisible loops. The total work done by all sources (which isnot an exact differential) is then

δW =∑k,j

Vkj dqj . (91)

For nondissipative circuits all this work goes into thevariation of the internal energy dE, which can be computedfrom T + U by subtracting the contribution from the voltagesources. The work-energy theorem in this case thus reads

δW = dE . (92)

In the presence of current sources the work done is

δW =∑k,j

Ikj (dφk − dφj ), (93)

where Ikj is the current of the source between the kth and j thjunctions (or 0 if none such source exists) and dφk − dφj isthe difference in flux on the two sides of the source. This workaccounts for the change dE of internal energy which derivesfrom T + U by subtracting the contribution from the currentsources to give a balance formally equal to Eq. (92).

C. Generalized Joule’s first law

In the dissipative case, on the other hand, we need totake into account the part of the work done by the voltagesources that goes into a “generalized heat” which accountsfor the heat generated in the resistances (if present) and the“heat” generated from the dissipative components of the statevariables.

Mathematically, this amounts to

δW − dE =∑

j

∂H∂qj

qj dt +∑

i

∂H∂yi

yidt (94)

for voltage-controlled circuits. On the other hand, in thepresence of current sources, Eq. (94) needs to be changedinto

δW − dE =∑

j

∂H∂φj

φj dt +∑

i

∂H∂yi

yidt. (95)

Equations (94) and (95) are the generalized Joule’s first lawsfor dissipative systems and are important yardsticks, togetherwith the Euler-Lagrange EOMs, to test the validity of agiven Lagrangian formulation. We note, in particular, theidentification of the energy loss due to memory, given by thelast terms on the right-hand side of Eqs. (94) and (95), whichare not present in the formulation of standard circuit elements.

VIII. QUANTIZATION

Having shown the Hamiltonian formulation for classicalnondissipative circuits, we now embark on the quantizationof these Hamiltonians, which will be of importance atlow temperatures and mesoscopic/nanoscopic length scales.

Instead of proceeding with a general circuit, in this case wefind it more instructive to first work out explicit examples.We consider first a voltage source connected in series witha memcapacitive system and a meminductive system. Thenwe look at a current source connected in parallel with thesesystems. These two circuits can be realized experimentally(see, e.g., Ref. 35) and are therefore ideal test beds for theconcept of memory quanta, namely, quantized excitations ofthe memory degrees of freedom of these circuits.

A. Example: series LC circuit

We consider a voltage source connected in series with amemcapacitive system and a meminductive one as depictedin Fig. 9(a). The case of such a circuit with no memory wasquantized in previous works26,36 and will be generalized here.The Hamiltonian for this circuit is found using Eqs. (7) and(35) for the memcapacitive system and Eqs. (50) and (8) forthe meminductive one and reads

H = T + U = 1

2

∑i

c−11i z2

1i + 1

2

∑i

c−12i z2

2i

+ q2

2C(y1,V1,t)+ φ2

2L(y2,φ,t)+ U1(y1,V1,t)

+ U2(y2,φ,t) − qV (t), (96)

where the index 1 corresponds to the capacitor and the index2 to the inductor. yji is the ith memory coordinate of thej th memory element, and zji is its canonically conjugatemomentum as defined in Eq. (87). q is the charge flowingthrough the circuit, and φ is the flux on the inductor. V1 is thevoltage on the capacitor, and V (t) is the voltage of the source.

Under reasonable assumption of stability of the values ofy1 and y2, we expand U1(y1,V1,t) and U2(y2,φ2,t) at theirminima with respect to y1 and y2, respectively. Several suchminima can exist for y1 or y2 in certain memory elements.37 Inthis case we should choose one minimum based on the initialconditions. The definitions of yi are shifted by constants tomake them zero at their respective minima. This shift doesnot affect the form of the other terms in the Hamiltonian. Inaddition, we define (C−1)(y1,V1,t) and (L−1)(y2,φ,t) by

(C−1)(y1,V1,t) ≡ 1

C(y1,V1,t)− 1

C0(V1,t), (97)

(L−1)(y2,φ,t) ≡ 1

L(y2,φ,t)− 1

L0(φ,t), (98)

where for brevity we have defined C0(V1,t) ≡ C(0,V1,t) andL0(p,t) ≡ L(0,p,t).

(a) (b)

FIG. 9. (a) Schematic of a series voltage-controlled memoryelement LC circuit. (b) Schematic of a parallel current-controlledmemory element LC circuit.

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LAGRANGE FORMALISM OF MEMORY CIRCUIT . . . PHYSICAL REVIEW B 85, 165428 (2012)

Discarding constant terms in Ui and neglecting higherorder terms in xi , we can write Ui ≈ ∑

j djiy2ji/2, and the

Hamiltonian takes the form

H = Hq + Hx + Hint, (99)

Hq = φ2

2L0+ q2

2C0− qV (t), (100)

Hy = 1

2

∑ij

c−1ji z2

ji + 1

2

∑ij

djiy2ji , (101)

Hint = (L−1)φ2

2+ (C−1)q2

2, (102)

where the Hamiltonian was divided into the “charge” part Hq ,the “memory” part Hy , and the “interaction” part Hint.

We next introduce the bosonic creation and annihilationoperators defined by

a =√

L0ω0

2h

(q + iφ

L0ω0

), (103)

a† =√

L0ω0

2h

(q − iφ

L0ω0

), (104)

bji =√

cjiωji

2h

(yji + izji

cjiωji

), (105)

b†ji =

√cjiωji

2h

(yji − izji

cjiωji

), (106)

where a† and a (b†ji and bji) create and destroy charge(memory) quanta, respectively.

The frequency of the charge oscillator, ω0, is the circuitresonance frequency, (L0C0)−1/2, while the frequencies of thememory quanta oscillators are analogously given by

ωji ≡√

dji

cji

. (107)

Plugging these relations into the Hamiltonian in Eq. (99) finallygives the quantized form:

Hq = hω0

(a†a + 1

2

)−

√h

2L0ω0(a + a†)V (t), (108)

Hx =∑ij

hωji

(b†jibji + 1

2

), (109)

Hint = −hL0ω0

4 (L−1)(y2,φ,t)(a − a†)2

+ h

4L0ω0 (C−1)(y1,V1,t)(a + a†)2, (110)

where V1 is quantized by solving the equation V1 =q/C(y1,V1,t) to give V1 = V1(y1,q,t), which translates toV1 = V1(y1 ,q,t) after quantization.

It is now clearly seen that Hq includes only terms cor-responding to the charge quanta, while Hy includes thosecorresponding to memory. Hint couples the two quanta witha coupling term that has at least three ladder operators.

A simple example18 that illustrates this result is a circuitinvolving a normal inductor of inductance L in series with amemcapacitor that has its upper plate of mass m hanging ona spring with spring constant k (Fig. 10). This memcapacitorcould be a representation of, e.g., a nanoelectromechanical

FIG. 10. Elastic memcapacitive system connected to a normalinductor.

system.38–40 If the displacement of the upper plate from itsequilibrium position is denoted by y and its distance from thelower plate at this position is d0, the capacitance can be easilyseen to be given by18

C(x) = C0

1 + y/d0, (111)

where C0 is the capacitance at equilibrium. Using the aboveformalism to quantize the Hamiltonian and keeping onlyenergy-conserving terms, we find Eq. (110) is reduced to

Hint = h3/2

4√

2d−1

0 (LC0)−1/2(mk)−1/4(a†2b + b†a2). (112)

This type of interaction is of the same kind as the oneencountered in the quantum treatment of second-harmonicgeneration in optics.41

If the circuit can be built to satisfy (k/m)1/2 = 2(LC0)−1/2

the interaction will produce a splitting of the degeneracy ofthe levels that is of first order in Hint and which may belarge enough to be detected experimentally. (See also theConclusions for an order of magnitude estimate of when toexpect quantum effects to dominate.)

B. Example: parallel LC circuit

As a second example let us consider a parallel memory LC

circuit as plotted in Fig. 9(b). In this circuit a current sourceis connected in parallel with a memcapacitive system and ameminductive system. The Hamiltonian for this system can befound utilizing Eqs. (39) and (40) for the former and Eqs. (56)and (57) for the latter, resulting in

H =T + U=1

2

∑i

c−11i z2

1i +1

2

∑i

c−12i z2

2i +q2

2C(y1,φ,t)

+ φ2

2L(y2,I2,t)+ U1(y1,C

−1φ,t)+U2(y2,LI2,t)−φI (t),

(113)

with the same definitions as in Eq. (96), except for φ being theflux in the inductor and q being the charge on the capacitor.I2 is the current through the inductor and I (t) is the current ofthe source.

Proceeding in a completely analogous way to the previoussubsection with the definitions of the ladder operators inEqs. (103)–(106) modified by the substitutions L0 → C0 andq → φ, we find the quantized Hamiltonian of this system tobe

H = Hq + Hy + Hint, (114)

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COHEN, PERSHIN, AND DI VENTRA PHYSICAL REVIEW B 85, 165428 (2012)

Hq = hω0

(a†a + 1

2

)−

√h

2C0ω0(a + a†)I (t), (115)

Hy =∑ij

hωji

(b†jibji + 1

2

), (116)

Hint = −hC0ω0

4 (L−1)(x2,I2,t)(a − a†)2

+ h

4C0ω0 (C−1)(x1,q,t)(a + a†)2, (117)

where I2 is quantized by solving the equation I2 =φ/L(y2,I2,t) to give I2 = I2(y2,φ,t), which reduces to I2 =I2(y2,φ,t) after quantization. This Hamiltonian is very similarto the one obtained for the series LC circuit. We will now showthese two Hamiltonians to be the basic building blocks for thequantized Hamiltonian of a general circuit with nondissipativeelements.

C. General circuit

We now proceed to find the quantized Hamiltonian for ageneral circuit network of memcapacitive systems, meminduc-tive systems, and voltage sources. Such a circuit can be dividedinto indivisible loops, each with charge qk as noted in Sec. VI.Using the methods of that section to find T and U , one canwrite the Hamiltonian for the network as H = T + U , which,after the substitution of qk and φk with ladder operators usingEqs. (103)–(106), reduces, apart from the interaction part, toa bilinear combination of them which is known to be exactlydiagonalizable by, e.g., a linear canonical transformation.

With regard to the quantization of the Hamiltonian ofcurrent-controlled circuits, the process is similar. We denotethe flux in each junction of the network with φk with acorresponding qk being the charge in the junction. Using themethods of Sec. VI, we write the Hamiltonian H = T + V

and then quantize it by writing φk and pk in terms of ladderoperators using the transformation from the previous subsec-tion. As in the voltage-controlled case, the noninteracting partof the Hamiltonian is again bilinear and can be diagonalized.

IX. CONCLUSIONS

To summarize, in this work we introduced the generalLagrangian formulation for the three basic memoryelements—memristive, memcapacitive, and meminductivesystems—and defined a fourth memory element, a mutualmeminductive system, for which we also gave the Lagrangeformalism. We showed how to write the Lagrangian for

a general circuit, including one with current sources. Theexamples given for the Lagrangian formalism demonstratedthat writing the Lagrangian and dissipation potential shouldbe the preferred choice for finding the EOMs of large memoryelement networks.

The Hamiltonian formalism for electric circuits was alsogeneralized to include memory, although only for nondissi-pative elements. As in previous works,26 we have found thatthe canonically conjugate momentum of charge is the fluxand vice versa. The generalized Joule’s first law was givenfor general circuits including ones with memory elements.This law can be used to verify the correctness of a givenLagrangian formulation. Lastly, we presented a scheme forthe quantization of a general nondissipative memory elementcircuit.

The quantum treatment of memory elements, and inparticular the example given in the text of a memcapacitorin series with an inductor (Fig. 10), begs the question ofunder which conditions one can measure quantum effects inthese systems. For quantum effects to be easily measurable,both the thermal fluctuation energy and the width of theenergy levels should be smaller than the oscillator energyquantum,26 i.e., kBT hω0 and Q 1, where Q = ω0R/L

is the quality factor of the oscillator, R is the loop resistance,and L is the inductance. Possible values for the capacitance andinductance in mesoscopic circuits can be taken to be 10−15 F42

and 10−10 H,43 respectively. If one assumes a temperatureof T = 20 mK and circuit resistance of 10 � or less, bothconditions mentioned above are satisfied. As noted for theexample above, the degeneracy condition, satisfiable by amemcapacitor,44 will lead to an experimentally detectablesplitting of the degenerate energy levels as a result of theinteraction between the memory quanta and charge quanta.

Future research in this field may include extending theHamiltonian formalism to dissipative circuits. One way todo this is, e.g., via a path-integral formulation45 of memoryelements. Along a parallel line, we expect the Lagrangianformalism discussed here to be of great value in the analysis ofcomplex networks with memory, which offer both fundamentaland applied research opportunities.

ACKNOWLEDGMENTS

M. D. is grateful to the Scuola Normale Superiore of Pisa,for the hospitality during a visit where part of this work waswritten, and to S. Pugnetti and R. Fazio for useful discussions.This work has been partially funded by the National ScienceFoundation Grant No. DMR-0802830.

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